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Solved Problems-22

Problems-22
Find the response current of a series RLC circuit consisting of a resistor R = 2Ω, an inductor L = 1H and a capacitor C = Description: Description: 10007.pngF when each of the following driving force voltage is applied:
  1. ramp voltage Description: Description: 10015.png,
  2. step voltage Description: Description: 10021.png,
  3. impulse voltage Description: Description: 10027.png, and
  4. doublet voltage Description: Description: 10033.png.
Solution
Applying KVL for the series RLC circuit we get,
 
Description: Description: 10039.png
  1. Ramp voltage Description: Description: 10049.png
When Description: Description: 10059.png
 
Description: Description: 10068.png
 
 
Taking Laplace transform,
Description: Description: 10074.png
 
Description: Description: 10080.png
 
Taking inverse Laplace transform, we, get,
Description: Description: 10086.png
  1. Step voltage Description: Description: 10092.png
When Description: Description: 10099.png
 
Description: Description: 10105.png
 
Taking Laplace transform,
Description: Description: 10114.png
 
Description: Description: 10120.png
 
Taking inverse Laplace transform, we, get,
Description: Description: 10132.png
  1. Impulse voltage Description: Description: 10138.png
When Description: Description: 10144.png
Description: Description: 10150.png
 
Taking Laplace transform,
Description: Description: 10156.png
Description: Description: 10163.png
∴ Description: Description: 10169.png
∴ Description: Description: 10177.png
  
Taking inverse Laplace transform, we, get,
Description: Description: 10193.png
  1. Doublet voltage Description: Description: 10199.png
When Description: Description: 10208.png
 
Description: Description: 10214.png
 
Taking Laplace transform,
Description: Description: 10221.png
⇒ Description: Description: 10228.png
 
Let, Description: Description: 10234.png
∴ Description: Description: 10240.png
∴ Description: Description: 10246.png
∴ 
Taking inverse Laplace transform, we, get,
 
Description: Description: 10263.png
 




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